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Department of Applied Mathematics and Theoretical Physics

  • 2016 - date: PhD student, DAMTP, CCA, University of Cambridge. 
  • 2015 - 2016: M.Math. (Part III), University of Cambridge.
  • 2012 - 2015: B.A. Maths, University of Cambridge.


Matthew is a member of the Applied Functional and Harmonic Analysis group within the Cambridge Centre for Analysis at DAMTP. Details of his research can be found here. His current interests are:

  • Spectral theory, in particular the existence (or non-existence!) of algorithms to solve computational problems.
  • Deep learning, its mathematical foundations and the problem of stability.
  • Spectral/pseudospectral methods and linear algebra in infinite dimensions.
  • Scattering problems and other problems related to computational fluid mechanics.


A full list of Matthew's publications can be seen here. Below is a list updated September 2019.

Journal Articles

(*denotes first authors where applicable, others alphabetical)

(paper prizes shown in orange)

Computational Spectral Theory

Pseudoergodic operators and periodic boundary conditions

Matthew Colbrook*

Mathematics of Computation, 2019.

preprint journal link
How to compute spectra with error control

(Smith-Knight/Rayleigh-Knight Prize Class I 2018)

Matthew Colbrook*, Bogdan Roman, Anders Hansen

Physical Review Letters, 2019 (chosen as cover article)

preprint journal link
On the infinite-dimensional QR algorithm

Matthew Colbrook*, Anders Hansen

Numerische Mathematik, 2019.

preprint journal link

Spectral Methods

A hybrid analytical-numerical technique for elliptic PDEs

Matthew Colbrook*, Thanasis Fokas, Parham Hashemzadeh

SIAM Journal on Scientific Computing (SISC), 2019.

preprint journal link
Extending the unified transform: curvilinear polygons and variable coefficient PDEs

Matthew Colbrook*

IMA Journal of Numerical Analysis, 2018.

preprint journal link
On the Fokas method for the solution of elliptic problems in both convex and non-convex polygonal domains

Matthew Colbrook*, Natasha Flyer, Bengt Fornberg

Journal of Computational Physics, 2018.

preprint journal link

Scattering Problems and Computational Fluid Mechanics

A spectral collocation method for acoustic scattering by multiple elastic plates

Matthew Colbrook*, Lorna Ayton

Journal of Sound and Vibration, 2019.

preprint journal link
The unified transform for mixed boundary condition problems in unbounded domains

(IMA Lighthill-Thwaites 2nd Prize 2019)

Matthew Colbrook*, Lorna Ayton, Athanassios Fokas

Proceedings of the Royal Society A, 2019.

preprint journal link
Scaling laws of passive-scalar diffusion in the interstellar medium

Matthew Colbrook*, Xiangcheng Ma, Philip Hopkins, Jonathan Squire

Monthly Notices of the Royal Astronomical Society, 2017.

preprint journal link


A hybrid analytical-numerical method for solving advection-dispersion problems on a half-line

Felipe de Barros, Matthew Colbrook, Athanassios Fokas

International Journal of Heat and Mass Transfer, 2019.

preprint journal link
Computing eigenvalues and eigenfunctions of the Laplacian for convex polygons

Matthew Colbrook*, Athanassios Fokas

Applied Numerical Mathematics, 2018.

preprint journal link

Refereed Conference Articles

The unified transform: a spectral collocation method for acoustic scattering

Lorna Ayton, Matthew Colbrook, Athanassios Fokas

AIAA/CEAS Aeroacoustics, 2019.

preprint journal link
On the existence of stable and accurate neural networks for image reconstruction

Matthew Colbrook*, Vegard Antun, Anders Hansen

Signal Processing with Adaptive Sparse Structured Representations (SPARS), 2019.


Submitted Articles/Preprints

Computing spectral measures and spectral types: new algorithms and classifications

Matthew Colbrook*

arXiv preprint
The foundations of spectral computations via the solvability complexity index hierarchy: Part I

Matthew Colbrook*, Anders Hansen

arXiv preprint
The foundations of spectral computations via the solvability complexity index hierarchy: Part II

Matthew Colbrook*

arXiv preprint
On the solvability complexity index hierarchy and towers of algorithms

Jonathan Ben-Artzi, Matthew Colbrook, Anders Hansen, Olavi Nevanlinna, Markus Seidel

arXiv preprint
Kernel density estimation with linked boundary conditions

Matthew Colbrook*, Zdravko Botev, Karsten Kuritz, Shev MacNamara

arXiv preprint
Fast and spectrally accurate numerical methods for perforated screens

(with applications to Robin boundary conditions)

Matthew Colbrook, Matthew Priddin

Stable and accurate neural networks for image reconstruction

Matthew Colbrook*

In prep.